I would like to know if there exists a formula to compute the $L^2$-Wasserstein distance between the laws $P_1$ and $P_2$ in path space of two diffusion processes: $$dx_t=f_1(x_t)dt + \sigma_1(x_t)dB^1_t$$ $$dy_t=f_2(y_t)dt + \sigma_2(y_t)dB^2_t$$ that is:

$$W(P_1,P_2):=\inf_{couplings} \mathbb{E}\left[\int_0^T |x_t - y_t|^2dt\right]^{1/2}$$ if my understanding of the Wasserstein metric is correct. By formula, I mean something that would depend only on functions $f_1,f_2,\sigma_1,\sigma_2$ and not on the processes $x_t$ and $y_t$.

As an example it is easy to compute it in the case $f_1=f_2=0$, $\sigma_1=1,\sigma_2=2$ for instance (write two correlated Brownian motions with correlation $\rho$ and minimize over $\rho$). In the general case I am lost. Any idea?

  • $\begingroup$ Are you familiar with the Brenier theorem for existence of optimal transport plans for the $L^2$ Wasserstein metric? This may be useful. $\endgroup$ – Dorian Jun 5 '12 at 19:00
  • $\begingroup$ @Dorian: No, I am not. How could it be useful? $\endgroup$ – user16215 Jun 8 '12 at 9:54
  • $\begingroup$ Explicit formulas may be too much to hope for in general, but you can certainly bound the distance, at least if the $f_i,\sigma_i$ are Lipschitz. Solve the SDEs strongly, with respect to the same Brownian motion, and then use the inequality $W(P_1,P_2)^2 \le \mathbb{E}[\int_0^T(X_t - Y_t)^2dt]$ along with standard stability estimates. $\endgroup$ – Dan Mar 1 '13 at 14:06
  • $\begingroup$ yes Dan, that's a good idea, thanks. $\endgroup$ – user16215 Mar 5 '13 at 17:26
  • $\begingroup$ I am also struggling with this question, have you made some progress on this problem? In my case the functions $f_1, f_2$ are even the same, $f_1=f_2$. Do you see a way to estimate the Wasserstein metric in terms of some norm of $\sigma_1-\sigma_2$? $\endgroup$ – Abakus Aug 15 '14 at 8:18

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